In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete normed vector space, possibly of infinite dimension.
Suppose that for each point x ∈ M, the fibre Ex = π−1(x) has been given the structure of a Banach space.
The collection of all Banach bundles can be made into a category by defining appropriate morphisms.
These two morphisms are required to satisfy two conditions (again, the second one is redundant in the finite-dimensional case): One can take a Banach bundle over one manifold and use the pull-back construction to define a new Banach bundle on a second manifold.
Specifically, let π : E → N be a Banach bundle and f : M → N a differentiable map (as usual, everything is Cp).